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I would like to determine the point $C$ in this image:
(assume I have radius value). After the hours of research and refreshing some memories from school days, I've got: Please assume... Point: (xCoordinate, yCoordinate): A (center): $(x_0,y_0)$ B: $(x_1,y_1) $ Plugging them into equation of circle $$x^2 + \left(y_0 + \left( \frac{y_1-y_0}{x_1-x_0}\right) (x-x_0)\right)^2 = r^2$$ Now how can I solve for $x$? I would like to have the equation that starts with "$x=$". Thanks a bunch in advance. |
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Because the point $C=(x,y)$ is lies on a line through $(x_0,y_0)$ and $(x_1,y_1)$, it is of the form $$(x,y)=(x_0+t(x_1-x_0),y_0+t(y_1-y_0))$$ for some real number $t$. In order for $C$ to lie on the circle, we must have that $$(t(x_1-x_0))^2+(t(y_1-y_0))^2=r^2,$$ and hence $$t=\frac{r}{\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}}.$$ Thus, the $x$-coordinate of $C$ is $$x=x_0+\frac{r(x_1-x_0)}{\sqrt{(x_1-x_0)^2+(y_1-y_0)^2}}.$$ |
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Hint: Assuming the coordinates of unknown point are $(X,Y)$, solve these two equations simultaneously: $$(Y-y_0)^2+(X-x_0)^2=r^2,~~~Y=\frac{y_1-y_0}{x_1-x_0}(X-x_0)-y_0$$ |
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